Interpolation that leads to the narrowest intervals and its application to expert systems and intelligent control

In many real-life situations, we want to reconstruct the dependencyy=f(x ₁,…, x_n) from the known experimental resultsx _i ^(k) , y^(k). In other words, we want tointerpolate the functionf from its known valuesy ^(k)=f(x ₁ ^(k) ,…, x _n ^(k) ) in finitely many points $\bar x^{(k)} = (x_1^{(k)} , \ldots ,x_n^{(k)} )$ , 1≤k≤N There are many functions that go through given points. How to choose one of them? The main goal of findingf is to be able to predicty based onx _i. If we getx _i from measurements, then usually, we only getintervals that containx _i. As a result of applyingf, we get an interval y of possible values ofy. It is reasonable to choosef for which the resulting interval is the narrowest possible. In this paper, we formulate this choice problem in mathematical terms, solve the corresponding problem for several simple cases, and describe the application of these solutions to intelligent control.     

Erzeugende Funktionen bei Spline-Interpolation mit äquidistanten Knoten

Using generating functions we obtain in the case ofn+1 equidistant data points a method for the calculation of the interpolating spline functions(x) of degree 2k+1 with boundary conditionss ^(κ) (x₀)=y ₀ ^(κ) ,s ^(κ) (x _n )=y _n ^(κ) , κ=1(1)k, which only needs the inversion of a matrix of orderk. The applicability of our method in the case of general boundary conditions is also mentioned.     

A Mixed DG Method for Linearized Incompressible Magnetohydrodynamics

We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous ? _k ³ ?? _k?1 elements whereas the magnetic part of the equations is approximated by discontinuous ? _k ³ ?? _k+1 elements. We carry out a complete a-priori error analysis of the method and prove that the energy norm error is convergent of order k in the mesh size. These results are verified in a series of numerical experiments.     

Solving the Euclidean bottleneck matching problem byk-relative neighborhood graphs

Given a set of pointsV in the plane, the Euclidean bottleneck matching problem is to match each point with some other point such that the longest Euclidean distance between matched points, resulting from this matching, is minimized. To solve this problem, we definek-relative neighborhood graphs, (kRNG) which are derived from Toussaint's relative neighborhood graphs (RNG). Two points are calledk-relative neighbors if and only if there are less thank points ofV which are closer to both of the two points than the two points are to each other. AkRNG is an undirected graph (V,E _r ^k ) whereE _r ^k is the set of pairs of points ofV which arek-relative neighbors. We prove that there exists an optimal solution of the Euclidean bottleneck matching problem which is a subset ofE _r ¹⁷ . We also prove that ¦E _r ^k ¦ < 18kn wheren is the number of points in setV. Our algorithm would construct a 17RNG first. This takesO(n ²) time. We then use Gabow and Tarjan's bottleneck maximum cardinality matching algorithm for general graphs whose time-complexity isO((n logn)^0.5 m), wherem is the number of edges in the graph, to solve the bottleneck maximum cardinality matching problem in the 17RNG. This takesO(n ^1.5 log^0.5 n) time. The total time-complexity of our algorithm for the Euclidean bottleneck matching problem isO(n ² +n ^1.5 log^0.5 n).     

Complexity-Theoretic Hierarchies Induced by Fragments of Gödel’s T

We introduce two hierarchies of unknown ordinal height. The hierarchies are induced by natural fragments of a calculus based on finite types and Gödel’s T, and all the classes in the hierarchies are uniformly defined without referring to explicit bounds. Deterministic complexity classes like logspace, p, pspace, linspace and exp are captured by the hierarchies. Typical subrecursive classes are also captured, e.g. the small relational Grzegorczyk classes ? _* ⁰ , ? _* ¹ and ? _* ² .     

Contracting Graphs to Paths and Trees

Vertex deletion and edge deletion problems play a central role in parameterized complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain a tree or a path, respectively, by a sequence of at most k edge contractions in G. For Tree Contraction, we present a randomized 4 ^k ? n ^O(1) time polynomial-space algorithm, as well as a deterministic 4.98 ^k ? n ^O(1) time algorithm, based on a variant of the color coding technique of Alon, Yuster and Zwick. We also present a deterministic 2 ^k+o(k)+n ^O(1) time algorithm for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k+3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ? coNP/poly. We find the latter result surprising because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with 4k ² vertices.     

A Simple D 2-Sampling Based PTAS for k-Means and Other Clustering Problems

Given a set of points \(P \subset\mathbb{R}^{d}\) , the k-means clustering problem is to find a set of k centers \(C = \{ c_{1},\ldots,c_{k}\}, c_{i} \in\mathbb{R}^{d}\) , such that the objective function ∑ _x∈P e(x,C)², where e(x,C) denotes the Euclidean distance between x and the closest center in C, is minimized. This is one of the most prominent objective functions that has been studied with respect to clustering. D ²-sampling (Arthur and Vassilvitskii, Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1027–1035, SIAM, Philadelphia, 2007) is a simple non-uniform sampling technique for choosing points from a set of points. It works as follows: given a set of points \(P \subset\mathbb{R}^{d}\) , the first point is chosen uniformly at random from P. Subsequently, a point from P is chosen as the next sample with probability proportional to the square of the distance of this point to the nearest previously sampled point. D ²-sampling has been shown to have nice properties with respect to the k-means clustering problem. Arthur and Vassilvitskii (Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1027–1035, SIAM, Philadelphia, 2007) show that k points chosen as centers from P using D ²-sampling give an O(logk) approximation in expectation. Ailon et al. (NIPS, pp. 10–18, 2009) and Aggarwal et al. (Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 15–28, Springer, Berlin, 2009) extended results of Arthur and Vassilvitskii (Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’07, pp. 1027–1035, SIAM, Philadelphia, 2007) to show that O(k) points chosen as centers using D ²-sampling give an O(1) approximation to the k-means objective function with high probability. In this paper, we further demonstrate the power of D ²-sampling by giving a simple randomized (1+?)-approximation algorithm that uses the D ²-sampling in its core.     

Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm

Let ${\mathcal{B}}$ be a centrally symmetric convex polygon of ?² and ‖p?q‖ be the distance between two points p,q∈?² in the normed plane whose unit ball is ${\mathcal{B}}$ . For a set T of n points (terminals) in ?², a ${\mathcal{B}}$ -network on T is a network N(T)=(V,E) with the property that its edges are parallel to the directions of ${\mathcal{B}}$ and for every pair of terminals t _i and t _j , the network N(T) contains a shortest ${\mathcal{B}}$ -path between them, i.e., a path of length ‖t _i ?t _j ‖. A minimum ${\mathcal{B}}$ -network on T is a ${\mathcal{B}}$ -network of minimum possible length. The problem of finding minimum ${\mathcal{B}}$ -networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX’99) in the case when the unit ball ${\mathcal{B}}$ is a square (and hence the distance ‖p?q‖ is the l ₁ or the l _∞-distance between p and q) and it has been shown recently by Chin, Guo, and Sun (Symposium on Computational Geometry, pp. 393–402, 2009) to be strongly NP-complete. Several approximation algorithms (with factors 8, 4, 3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum ${\mathcal{B}}$ -network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper (Chepoi et al. in Theor. Comput. Sci. 390:56–69, 2008, and APPROX-RANDOM, pp. 40–51, 2005) and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem.     

Polynomoperatoren in C k [a,b]

Some estimations for the relative projection constantP(∏ _n+k ^k ,Csik[a,b]) are given. By constructing an associated polynomial operatorL _n :C ⁰[a,b]→∏ _n ⁰ to a given polynomial operatorH _n+k :C ^k [a,b]→∏ _n+k ^k we get a lower bound for the projection constant. An upper bound forP(∏ _n+k ^k ,C ^k [a,b]) is obtained by the determination of the norms of appropriate polynomial operatorsP _n+k :C ^k [a,b]→∏ _n+k ^k . Further we give a convergence property for the sequence (P _n+k ) _n∈?.     

Query-Efficient Locally Decodable Codes of Subexponential Length

A k-query locally decodable code (LDC) C : Σ ⁿ → Γ ^N encodes each message x into a codeword C(x) such that each symbol of x can be probabilistically recovered by querying only k coordinates of C(x), even after a constant fraction of the coordinates has been corrupted. Yekhanin (in J ACM 55:1–16, 2008) constructed a 3-query LDC of subexponential length, N = exp(exp(O(log n/log log n))), under the assumption that there are infinitely many Mersenne primes. Efremenko (in Proceedings of the 41st annual ACM symposium on theory of computing, ACM, New York, 2009) constructed a 3-query LDC of length ${N_{2}={\rm exp}({\rm exp} (O(\sqrt{\log n\log\log n})))}$ with no assumption, and a 2 ^r -query LDC of length ${N_{r}={\rm exp}({\rm exp}(O(\sqrt[r]{\log n(\log \log n)^{r-1}})))}$ , for every integer r ≥ 2. Itoh and Suzuki (in IEICE Trans Inform Syst E93-D 2:263–270, 2010) gave a composition method in Efremenko’s framework and constructed a 3 · 2 ^r-2-query LDC of length N _r , for every integer r ≥ 4, which improved the query complexity of Efremenko’s LDC of the same length by a factor of 3/4. The main ingredient of Efremenko’s construction is the Grolmusz construction for super-polynomial size set-systems with restricted intersections, over ${\mathbb{Z}_m}$ , where m possesses a certain “good” algebraic property (related to the “algebraic niceness” property of Yekhanin in J ACM 55:1–16, 2008). Efremenko constructed a 3-query LDC based on m = 511 and left as an open problem to find other numbers that offer the same property for LDC constructions. In this paper, we develop the algebraic theory behind the constructions of Yekhanin (in J ACM 55:1–16, 2008) and Efremenko (in Proceedings of the 41st annual ACM symposium on theory of computing, ACM, New York, 2009), in an attempt to understand the “algebraic niceness” phenomenon in ${\mathbb{Z}_m}$ . We show that every integer m =  pq = 2 ^t ?1, where p, q, and t are prime, possesses the same good algebraic property as m = 511 that allows savings in query complexity. We identify 50 numbers of this form by computer search, which together with 511, are then applied to gain improvements on query complexity via Itoh and Suzuki’s composition method. More precisely, we construct a ${3^{\lceil r/2\rceil}}$ -query LDC for every positive integer r < 104 and a ${\left\lfloor (3/4)^{51} \cdot 2^{r}\right\rfloor}$ -query LDC for every integer r ≥ 104, both of length N _r , improving the 2 ^r queries used by Efremenko (in Proceedings of the 41st annual ACM symposium on theory of computing, ACM, New York, 2009) and 3 · 2 ^r-2 queries used by Itoh and Suzuki (in IEICE Trans Inform Syst E93-D 2:263–270, 2010). We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.     

Multi-letter quantum finite automata: decidability of the equivalence and minimization of states

Multi-letter quantum finite automata (QFAs) can be thought of quantum variants of the one-way multi-head finite automata (Hromkovi?, Acta Informatica 19:377?C384, 1983). It has been shown that this new one-way QFAs (multi-letter QFAs) can accept with no error some regular languages, for example (a?+?b)*b, that are not acceptable by QFAs of Moore and Crutchfield (Theor Comput Sci 237:275?C306, 2000) as well as Kondacs and Watrous (66?C75, 1997; Observe that 1-letter QFAs are exactly measure-once QFAs (MO-1QFAs) of Moore and Crutchfield (Theor Comput Sci 237:275?C306, 2000)). In this paper, we study the decidability of the equivalence and minimization problems of multi-letter QFAs. Three new results presented in this paper are the following ones: (1) Given a k ₁-letter QFA ${{\mathcal A}_1}$ and a k ₂-letter QFA ${{\mathcal A}_2}$ over the same input alphabet ??, they are equivalent if and only if they are (n ² m ^k-1?m ^k-1?+?k)-equivalent, where m =?|??| is the cardinality of ??, k =?max(k ₁,k ₂), and n =?n ₁?+?n ₂, with n ₁ and n ₂ being numbers of states of ${{\mathcal A}_{1}}$ and ${{\mathcal A}_{2}}$ , respectively. When k =?1, this result implies the decidability of equivalence of measure-once QFAs (Moore and Crutchfield in Theor Comput Sci 237:275?C306, 2000). (It is worth mentioning that our technical method is essentially different from the previous ones used in the literature.) (2) A polynomial-time O(m ^2k-1 n ⁸?+?km ^k n ⁶) algorithm is designed to determine the equivalence of any two multi-letter QFAs (see Theorems 2 and 3; Observe that if a brute force algorithm to determine equivalence would be used, as suggested by the decidability outcome of the point (1), the worst case time complexity would be exponential). Observe also that time complexity is expressed here in terms of the number of states of the multi-letter QFAs and k can be seen as a constant. (3) It is shown that the states minimization problem of multi-letter QFAs is solvable in EXPSPACE. This implies also that the state minimization problem of MO-1QFAs (see Moore and Crutchfield in Theor Comput Sci 237:275?C306, 2000, page 304, Problem 5), an open problem stated in that paper, is also solvable in EXPSPACE.     

On Uniformity and Circuit Lower Bounds

We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts:

1. Lower bounds against medium-uniform circuits. Informally, a circuit class is “medium uniform” if it can be generated by an algorithmic process that is somewhat complex (stronger than LOGTIME) but not infeasible. Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against medium-uniform circuit classes, including: ? For all k, P is not contained in P-uniform SIZE(n ^k ). That is, for all k, there is a language \({L_k \in {\textsf P}}\) that does not have O(n ^k )-size circuits constructible in polynomial time. This improves Kannan’s lower bound from 1982 that NP is not in P-uniform SIZE(n ^k ) for any fixed k. ? For all k, NP is not in \({{\textsf P}^{\textsf NP}_{||}-{\textsf {uniform SIZE}}(n^k)}\) .This also improves Kannan’s theorem, but in a different way: the uniformity condition on the circuits is stronger than that on the language itself. ? For all k, LOGSPACE does not have LOGSPACE-uniform branching programs of size n ^k .
2. Eliminating non-uniformity and (non-uniform) circuit lower bounds. We complement these results by showing how to convert any potential simulation of LOGTIME-uniform NC ¹ in ACC ⁰/poly or TC ⁰/poly into a medium-uniform simulation using small advice. This lemma can be used to simplify the proof that faster SAT algorithms imply NEXP circuit lower bounds and leads to the following new connection: ? Consider the following task: given a TC ⁰ circuit C of n ^O(1) size, output yes when C is unsatisfiable, and output no when C has at least 2 ^n-2 satisfying assignments. (Behavior on other inputs can be arbitrary.) Clearly, this problem can be solved efficiently using randomness. If this problem can be solved deterministically in 2 ^n-ω(log n) time, then \({{\textsf{NEXP}} \not \subset {\textsf{TC}}^0/{\rm poly}}\) .

Another application is to derandomize randomized TC ⁰ simulations of NC ¹ on almost all inputs: ?Suppose \({{\textsf{NC}}^1 \subseteq {\textsf{BPTC}}^0}\) . Then, for every ε > 0 and every language L in NC ¹, there is a LOGTIME?uniform TC ⁰ circuit family of polynomial size recognizing a language L′ such that L and L′ differ on at most \({2^{n^{\epsilon}}}\) inputs of length n, for all n.     

Approximating Node-Connectivity Augmentation Problems

We consider the (undirected) Node Connectivity Augmentation (NCA) problem: given a graph J=(V,E _J ) and connectivity requirements $\{r(u,v): u,v \in V\}$ , find a minimum size set I of new edges (any edge is allowed) such that the graph J∪I contains r(u,v) internally-disjoint uv-paths, for all u,v∈V. In Rooted NCA there is s∈V such that r(u,v)>0 implies u=s or v=s. For large values of k=max? _u,v∈V r(u,v), NCA is at least as hard to approximate as Label-Cover and thus it is unlikely to admit an approximation ratio polylogarithmic in k. Rooted NCA is at least as hard to approximate as Hitting-Set. The previously best approximation ratios for the problem were O(kln?n) for NCA and O(ln?n) for Rooted NCA. In this paper we give an approximation algorithm with ratios O(kln?² k) for NCA and O(ln?² k) for Rooted NCA. This is the first approximation algorithm with ratio independent of?n, and thus is a constant for any fixed k. Our algorithm is based on the following new structural result which is of independent interest. If $\mathcal{D}$ is a set of node pairs in a graph?J, then the maximum degree in the hypergraph formed by the inclusion minimal tight sets separating at least one pair in $\mathcal{D}$ is O(? ²), where ? is the maximum connectivity in J of a pair in $\mathcal{D}$ .     

Properties of interval operators

In this paper we give some properties of interval operatorsF which guarantee the convergence of the interval sequence {X _k} defined byX _k+1:=F(X_k)∩X_k to a unique fixed interval \(\hat X\) . This interval \(\hat X\) encloses the “zero-set”X ^* of a function strip \(G(x): = [g(x),\bar g(x)]\) . for some known interval operators we investigate under which assumptions these properties are valid.     

Zur Konvergenz der Ableitungen von Interpolationspolynomen

Letp _n denote the polynomial of degreen or less that interpolates a given smooth functionf at the ?eby?ev nodest _j ⁿ =cos(jπ/n), 0≤j≤n, and let ‖·‖ be the maximum norm inC[?1, 1]. It is proved that fork-th derivatives (2≤k≤n) estimates of the following type hold $$\parallel f^{(k)} - p_n^{(k)} \parallel \leqslant c_k n^{k - 1} \inf \{ \parallel f^{(k)} - q\parallel :q \in \Pi _{n - k} \} .$$ In this relationc _k only depends onk andΠ _n?k denotes the space of polynomials up to degreen?k.     

k-pairwise disjoint paths routing in perfect hierarchical hypercubes

Hierarchical hypercubes (HHC), also known as cube-connected cubes, have been introduced in the literature as an interconnection network for massively parallel systems. Effectively, they can connect a large number of nodes while retaining a small diameter and a low degree compared to a hypercube of the same size. Especially (2 ^m +m)-dimensional hierarchical hypercubes ( $\mathit {HHC}_{2^{m}+m}$ ), called perfect HHCs, are popular as they are symmetrical, which is a critical property when designing routing algorithms. In this paper, we describe an algorithm finding, in an $\mathit{HHC}_{2^{m}+m}$ , mutually node-disjoint paths connecting k=?(m+1)/2? pairs of distinct nodes. This problem is known as the k-pairwise disjoint-path routing problem and is one of the important routing problems when dealing with interconnection networks. In an $\mathit{HHC}_{2^{m}+m}$ , our algorithm finds paths of lengths at most 2 ^m+1+m(2 ^m+1+1)+4 in O(2^5m ) time, where 2 ^m+1 is the diameter of an $\mathit{HHC}_{2^{m}+m}$ . Also, we have shown through an experiment that, in practice, the lengths of the generated paths are significantly lower than the worst-case theoretical estimations.     

Amplifying circuit lower bounds against polynomial time, with applications

We give a self-reduction for the Circuit Evaluation problem (CircEval) and prove the following consequences.

1. Amplifying size–depth lower bounds. If CircEval has Boolean circuits of n ^k size and n ^1?δ depth for some k and δ, then for every ${\epsilon > 0}$ , there is a δ′ > 0 such that CircEval has circuits of ${n^{1 + \epsilon}}$ size and ${n^{1- \delta^{\prime}}}$ depth. Moreover, the resulting circuits require only ${\tilde{O}(n^{\epsilon})}$ bits of non-uniformity to construct. As a consequence, strong enough depth lower bounds for Circuit Evaluation imply a full separation of P and NC (even with a weak size lower bound).
2. Lower bounds for quantified Boolean formulas. Let c, d > 1 and e < 1 satisfy c < (1 ? e + d )/d. Either the problem of recognizing valid quantified Boolean formulas (QBF) is not solvable in TIME[n ^c ], or the Circuit Evaluation problem cannot be solved with circuits of n ^d size and n ^e depth. This implies unconditional polynomial-time uniform circuit lower bounds for solving QBF. We also prove that QBF does not have n ^c -time uniform NC circuits, for all c < 2.

     

The method of interlineation of vector functions \vec{w} (x, y, z, t) on a system of vertical straight lines and its application in crosshole seismic tomography

The authors propose a method to construct interlineation operators for vector functions $ \vec{w} $ (x, y, z, t) on a system of arbitrarily located vertical straight lines. The method allows calculating the vector $ \vec{w} $ at each point (x, y, z) between straight lines Γ _k for any instant of time t ≥ 0. They are proposed to be used to construct a crosshole accelerometer to model Earth’s crust on the basis of seismic sounding data $ {{\vec{w}}_k}\left( {z,t} \right),\,k=\overline{1,M} $ , about the vector of acceleration $ \vec{w} $ (x, y, z, t) received by accelerometers at each chink Γ _k .     

The k-in-a-Path Problem for Claw-free Graphs

The k-in-a-Path problem is to test whether a graph contains an induced path spanning k given vertices. This problem is NP-complete in general graphs, already when k=3. We show how to solve it in polynomial time on claw-free graphs, when k is an arbitrary fixed integer not part of the input. As a consequence, also the k-Induced Disjoint Paths and the k-in-a-Cycle problem are solvable in polynomial time on claw-free graphs for any fixed k. The first problem has as input a graph G and k pairs of specified vertices (s _i ,t _i ) for i=1,…,k and is to test whether G contain k mutually induced paths P _i such that P _i connects s _i and t _i for i=1,…,k. The second problem is to test whether a graph contains an induced cycle spanning k given vertices. When k is part of the input, we show that all three problems are NP-complete, even for the class of line graphs, which form a subclass of the class of claw-free graphs.     

Eine stets quadratisch konvergente Modifikation des Steffensen-Verfahrens

It is shown that the following modification of the Steffensen procedurex _n+1=x _n ?k _s (x _n )f(x _n ) (f[x _n ,x _n ?f(x _n )])^?1 (n=0,1,...) withk _s (x)=(1?z _s (x))^?1,z _s (x)=f(x) 2f[x?f(x),x,x+f(x)]×(f[x,x?f(x)])^?2 is quadratically convergent to the root of the equation \(f(x) = (x - \bar x)^p g(x) = 0(p > 0,g(\bar x) \ne 0)\) . Furthermore \(\mathop {\lim }\limits_{n \to \infty } k_s (x_n ) = p\) holds.